Timeline for Does the lattice of coverings embed in the lattice of partitions?
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 23, 2016 at 7:29 | vote | accept | Dominic van der Zypen | ||
| Sep 23, 2016 at 0:34 | comment | added | Joel David Hamkins | That construction shows in ZFC that for any infinite set $X$, the number of proper covers of $X$ is $2^{2^{|X|}}$, but the number of partitions of $X$ is merely $2^{|X|}$, so there is no injection from proper covers of $X$ to partitions of $X$. | |
| Sep 22, 2016 at 15:20 | comment | added | Joel David Hamkins | Another way to get $2^{\frak{c}}$ many proper covers: divide $X$ into pairs, and then consider any covering consisting of those pairs, plus any collection of sets that select exactly one from each pair. These are all proper coverings, and there are $2^{\frak{c}}$ many of them. | |
| Sep 22, 2016 at 14:19 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
not a lattice
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| Sep 22, 2016 at 13:29 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |